for a particle trapped in a potential well with potential , with as . This ODE always admits global solutions from arbitrary initial positions and initial velocities , thanks to conservation of the Hamiltonian . As this Hamiltonian is coercive (in that its level sets are compact), solutions to this equation are always almost periodic. On the other hand, as can already be seen using the harmonic oscillator (and direct sums of this system), this equation can generate periodic solutions, as well as quasiperiodic solutions.

All quasiperiodic motions are almost periodic. However, there are many examples of dynamical systems that admit solutions that are almost periodic but not quasiperiodic. So one can pose the question: are the dynamics of potential wells universal in the sense that they can capture all almost periodic solutions?

A precise question can be phrased as follows. Let be a compact manifold, and let be a smooth vector field on ; to avoid degeneracies, let us take to be non-singular in the sense that it is everywhere non-vanishing. Then the trajectories of the first-order ODE

for are always global and almost periodic. Can we then find a (coercive) potential for some , as well as a smooth embedding , such that every solution to (2) pushes forward under to a solution to (1)? (Actually, for technical reasons it is preferable to map into the phase space , rather than position space , but let us ignore this detail for this discussion.)

It turns out that the answer is no; there is a very specific obstruction. Given a pair as above, define a strongly adapted -form to be a -form on such that is pointwise positive, and the Lie derivative is an exact -form. We then have

Theorem 1 A smooth compact non-singular dynamics can be embedded smoothly in a potential well system if and only if it admits a strongly adapted -form.

Interestingly, the same obstruction also works for potential wells in a more general Riemannian manifold than , or for nonlinear wave equations with a potential; combining the two, the obstruction is also present for wave maps with a potential.

It is then natural to ask whether this obstruction is non-trivial, in the sense that there are at least some examples of dynamics that do not support strongly adapted -forms (and hence cannot be modeled smoothly by the dynamics of a potential well, nonlinear wave equation, or wave maps). I posed this question on MathOverflow, and Robert Bryant provided a very nice construction, showing that the vector field on the -torus had no strongly adapted -forms, and hence the dynamics of this vector field cannot be smoothly reproduced by a potential well, nonlinear wave equation, or wave map:

On the other hand, the suspension of any diffeomorphism does support a strongly adapted -form (the derivative of the time coordinate), and using this and the previous theorem I was able to embed a universal Turing machine into a potential well. In particular, there are flows for an explicitly describable potential well whose trajectories have behavior that is undecidable using the usual ZFC axioms of set theory! So potential well dynamics are “effectively” universal, despite the presence of the aforementioned obstruction.

In my previous work on blowup for Navier-Stokes like equations, I speculated that if one could somehow replicate a universal Turing machine within the Euler equations, one could use this machine to create a “von Neumann machine” that replicated smaller versions of itself, which on iteration would lead to a finite time blowup. Now that such a mechanism is present in nonlinear wave equations, it is tempting to try to make this scheme work in that setting. Of course, in my previous paper I had already demonstrated finite time blowup, at least in a three-dimensional setting, but that was a relatively simple discretely self-similar blowup in which no computation occurred. This more complicated blowup scheme would be significantly more effort to set up, but would be proof-of-concept that the same scheme would in principle be possible for the Navier-Stokes equations, assuming somehow that one can embed a universal Turing machine into the Euler equations. (But I’m still hopelessly stuck on how to accomplish this latter task…)

where is the unknown field, and is the nonlinearity, which we assume to have all derivatives bounded. A typical example of such an equation is the higher-dimensional sine-Gordon equation

for a scalar field . Here is the d’Alembertian operator. We restrict attention here to classical (i.e. smooth) solutions to (1).

We do not assume any Hamiltonian structure, so we do not require to be a gradient of a potential . But even without such Hamiltonian structure, the equation (1) is very well behaved, with many a priori bounds available. For instance, if the initial position and initial velocity are smooth and compactly supported, then from finite speed of propagation has uniformly bounded compact support for all in a bounded interval. As the nonlinearity is bounded, this immediately places in in any bounded time interval, which by the energy inequality gives an a priori bound on in this time interval. Next, from the chain rule we have

which (from the assumption that is bounded) shows that is in , which by the energy inequality again now gives an a priori bound on .

One might expect that one could keep iterating this and obtain a priori bounds on in arbitrarily smooth norms. In low dimensions such as , this is a fairly easy task, since the above estimates and Sobolev embedding already place one in , and the nonlinear map is easily verified to preserve the space for any natural number , from which one obtains a priori bounds in any Sobolev space; from this and standard energy methods, one can then establish global regularity for this equation (that is to say, any smooth choice of initial data generates a global smooth solution). However, one starts running into trouble in higher dimensions, in which no bound is available. The main problem is that even a really nice nonlinearity such as is unbounded in higher Sobolev norms. The estimates

and

ensure that the map is bounded in low regularity spaces like or , but one already runs into trouble with the second derivative

where there is a troublesome lower order term of size which becomes difficult to control in higher dimensions, preventing the map to be bounded in . Ultimately, the issue here is that when is not controlled in , the function can oscillate at a much higher frequency than ; for instance, if is the one-dimensional wave for some and , then oscillates at frequency , but the function more or less oscillates at the larger frequency .

In medium dimensions, it is possible to use dispersive estimates for the wave equation (such as the famous Strichartz estimates) to overcome these problems. This line of inquiry was pursued (albeit for slightly different classes of nonlinearity than those considered here) by Heinz-von Wahl, Pecher (in a series of papers), Brenner, and Brenner-von Wahl; to cut a long story short, one of the conclusions of these papers was that one had global regularity for equations such as (1) in dimensions . (I reprove this result using modern Strichartz estimate and Littlewood-Paley techniques in an appendix to my paper. The references given also allow for some growth in the nonlinearity , but we will not detail the precise hypotheses used in these papers here.)

In my paper, I complement these positive results with an almost matching negative result:

Theorem 1 If and , then there exists a nonlinearity with all derivatives bounded, and a solution to (1) that is smooth at time zero, but develops a singularity in finite time.

The construction crucially relies on the ability to choose the nonlinearity , and also needs some injectivity properties on the solution (after making a symmetry reduction using an assumption of spherical symmetry to view as a function of variables rather than ) which restricts our counterexample to the case. Thus the model case of the higher-dimensional sine-Gordon equation is not covered by our arguments. Nevertheless (as with previous finite-timeblowup resultsdiscussed on this blog), one can view this result as a barrier to trying to prove regularity for equations such as in eleven and higher dimensions, as any such argument must somehow use a property of that equation that is not applicable to the more general system (1).

Let us first give some back-of-the-envelope calculations suggesting why there could be finite time blowup in eleven and higher dimensions. For sake of this discussion let us restrict attention to the sine-Gordon equation . The blowup ansatz we will use is as follows: for each frequency in a sequence of large quantities going to infinity, there will be a spacetime “cube” on which the solution oscillates with “amplitude” and “frequency” , where is an exponent to be chosen later; this ansatz is of course compatible with the uncertainty principle. Since as , this will create a singularity at the spacetime origin . To make this ansatz plausible, we wish to make the oscillation of on driven primarily by the forcing term at . Thus, by Duhamel’s formula, we expect a relation roughly of the form

on , where is the usual free wave propagator, and is the indicator function of .

On , oscillates with amplitude and frequency , we expect the derivative to be of size about , and so from the principle of stationary phase we expect to oscillate at frequency about . Since the wave propagator preserves frequencies, and is supposed to be of frequency on we are thus led to the requirement

Next, when restricted to frequencies of order , the propagator “behaves like” , where is the spherical averaging operator

where is surface measure on the unit sphere , and is the volume of that sphere. In our setting, is comparable to , and so we have the informal approximation

on .

Since is bounded, is bounded as well. This gives a (non-rigorous) upper bound

which when combined with our ansatz that has ampitude about on , gives the constraint

To turn this ansatz into an actual blowup example, we will construct as the sum of various functions that solve the wave equation with forcing term in , and which concentrate in with the amplitude and frequency indicated by the above heuristic analysis. The remaining task is to show that can be written in the form for some with all derivatives bounded. For this one needs some injectivity properties of (after imposing spherical symmetry to impose a dimensional reduction on the domain of from dimensions to ). This requires one to construct some solutions to the free wave equation that have some unusual restrictions on the range (for instance, we will need a solution taking values in the plane that avoid one quadrant of that plane). In order to do this we take advantage of the very explicit nature of the fundamental solution to the wave equation in odd dimensions (such as ), particularly under the assumption of spherical symmetry. Specifically, one can show that in odd dimension , any spherically symmetric function of the form

for an arbitrary smooth function , will solve the free wave equation; this is ultimately due to iterating the “ladder operator” identity

This precise and relatively simple formula for allows one to create “bespoke” solutions that obey various unusual properties, without too much difficulty.

It is not clear to me what to conjecture for . The blowup ansatz given above is a little inefficient, in that the frequency component of the solution is only generated from a portion of the component, namely the portion close to a certain light cone. In particular, the solution does not saturate the Strichartz estimates that are used to establish the positive results for , which helps explain the slight gap between the positive and negative results. It may be that a more complicated ansatz could work to give a negative result in ten dimensions; conversely, it is also possible that one could use more advanced estimates than the Strichartz estimate (that somehow capture the “thinness” of the fundamental solution, and not just its dispersive properties) to stretch the positive results to ten dimensions. Which side the case falls in all come down to some rather delicate numerology.

where is the unknown scalar field, is the d’Alambertian operator, and is an exponent. We can generalise this equation to the defocusing nonlinear wave system

where is now a system of scalar fields, and is a potential which is homogeneous of degree and strictly positive away from the origin; the scalar equation corresponds to the case where and . We will be interested in smooth solutions to (2). It is only natural to restrict to the smooth category when the potential is also smooth; unfortunately, if one requires to be homogeneous of order all the way down to the origin, then cannot be smooth unless it is identically zero or is an odd integer. This is too restrictive for us, so we will only require that be homogeneous away from the origin (e.g. outside the unit ball). In any event it is the behaviour of for large which will be decisive in understanding regularity or blowup for the equation (2).

Using this conserved energy, it is possible to establish global regularity for the Cauchy problem (2) in the energy-subcritical case when , or when and . This means that for any smooth initial position and initial velocity , there exists a (unique) smooth global solution to the equation (2) with and . These classical global regularity results (essentially due to Jörgens) were famously extended to the energy-critical case when and byGrillakis, Struwe, and Shatah-Struwe (though for various technical reasons, the global regularity component of these results was limited to the range ). A key tool used in the energy-critical theory is the Morawetz estimate

which can be proven by manipulating the properties of the stress-energy tensor

(with the usual summation conventions involving the Minkowski metric ) and in particular exploiting the divergence-free nature of this tensor: See for instance the text of Shatah-Struwe, or my own PDE book, for more details. The energy-critical regularity results have also been extended to slightly supercritical settings in which the potential grows by a logarithmic factor or so faster than the critical rate; see the results of myself and of Roy.

This leaves the question of global regularity for the energy supercritical case when and . On the one hand, global smooth solutions are known for small data (if vanishes to sufficiently high order at the origin, see e.g. the work of Lindblad and Sogge), and global weak solutions for large data were constructed long ago by Segal. On the other hand, the solution map, if it exists, is known to be extremely unstable, particularly at high frequencies; see for instance this paper of Lebeau, this paper of Christ, Colliander, and myself, this paper of Brenner and Kumlin, or this paper of Ibrahim, Majdoub, and Masmoudi for various formulations of this instability. In the case of the focusing NLW , one can easily create solutions that blow up in finite time by ODE constructions, for instance one can take with , which blows up as approaches . However the situation in the defocusing supercritical case is less clear. The strongest positive results are of Kenig-MerleandKillip-Visan, which show (under some additional technical hypotheses) that global regularity for such equations holds under the additional assumption that the critical Sobolev norm of the solution stays bounded. Roughly speaking, this shows that “Type II blowup” cannot occur for (2).

Our main result is that finite time blowup can in fact occur, at least for three-dimensional systems where the number of degrees of freedom is sufficiently large:

Theorem 1 Let , , and . Then there exists a smooth potential , positive and homogeneous of degree away from the origin, and a solution to (2) with smooth initial data that develops a singularity in finite time.

The rather large lower bound of on here is primarily due to our use of the Nash embedding theorem (which is the first time I have actually had to use this theorem in an application!). It can certainly be lowered, but unfortunately our methods do not seem to be able to bring all the way down to , so we do not directly exhibit finite time blowup for the scalar supercritical defocusing NLW. Nevertheless, this result presents a barrier to any attempt to prove global regularity for that equation, in that it must somehow use a property of the scalar equation which is not available for systems. It is likely that the methods can be adapted to higher dimensions than three, but we take advantage of some special structure to the equations in three dimensions (related to the strong Huygens principle) which does not seem to be available in higher dimensions.

The blowup will in fact be of discrete self-similar type in a backwards light cone, thus will obey a relation of the form

for some fixed (the exponent is mandated by dimensional analysis considerations). It would be natural to consider continuously self-similar solutions (in which the above relation holds for all, not just one ). And rough self-similar solutions have been constructed in the literature by perturbative methods (see this paper of Planchon, or this paper of Ribaud and Youssfi). However, it turns out that continuously self-similar solutions to a defocusing equation have to obey an additional monotonicity formula which causes them to not exist in three spatial dimensions; this argument is given in my paper. So we have to work just with discretely self-similar solutions.

Because of the discrete self-similarity, the finite time blowup solution will be “locally Type II” in the sense that scale-invariant norms inside the backwards light cone stay bounded as one approaches the singularity. But it will not be “globally Type II” in that scale-invariant norms stay bounded outside the light cone as well; indeed energy will leak from the light cone at every scale. This is consistent with the results of Kenig-MerleandKillip-Visan which preclude “globally Type II” blowup solutions to these equations in many cases.

We now sketch the arguments used to prove this theorem. Usually when studying the NLW, we think of the potential (and the initial data ) as being given in advance, and then try to solve for as an unknown field. However, in this problem we have the freedom to select . So we can look at this problem from a “backwards” direction: we first choose the field , and then fit the potential (and the initial data) to match that field.

Now, one cannot write down a completely arbitrary field and hope to find a potential obeying (2), as there are some constraints coming from the homogeneity of . Namely, from the Euler identity

Furthermore, taking a derivative of (3) we obtain another constraining equation

that does not explicitly involve the potential . Actually, one can write this equation in the more familiar form

where is the stress-energy tensor

now written in a manner that does not explicitly involve .

With this reformulation, this suggests a strategy for locating : first one selects a stress-energy tensor that is divergence-free and obeys suitable positive definiteness and self-similarity properties, and then locates a self-similar map from the backwards light cone to that has that stress-energy tensor (one also needs the map (or more precisely the direction component of that map) injective up to the discrete self-similarity, in order to define consistently). If the stress-energy tensor was replaced by the simpler “energy tensor”

then the question of constructing an (injective) map with the specified energy tensor is precisely the embedding problem that was famously solved by Nash (viewing as a Riemannian metric on the domain of , which in this case is a backwards light cone quotiented by a discrete self-similarity to make it compact). It turns out that one can adapt the Nash embedding theorem to also work with the stress-energy tensor as well (as long as one also specifies the mass density , and as long as a certain positive definiteness property, related to the positive semi-definiteness of Gram matrices, is obeyed). Here is where the dimension shows up:

Proposition 2 Let be a smooth compact Riemannian -manifold, and let . Then smoothly isometrically embeds into the sphere .

Proof: The Nash embedding theorem (in the form given in this ICM lecture of Gunther) shows that can be smoothly isometrically embedded into , and thus in for some large . Using an irrational slope, the interval can be smoothly isometrically embedded into the -torus , and so and hence can be smoothly embedded in . But from Pythagoras’ theorem, can be identified with a subset of for any , and the claim follows.

One can presumably improve upon the bound by being more efficient with the embeddings (e.g. by modifying the proof of Nash embedding to embed directly into a round sphere), but I did not try to optimise the bound here.

The remaining task is to construct the stress-energy tensor . One can reduce to tensors that are invariant with respect to rotations around the spatial origin, but this still leaves a fair amount of degrees of freedom (it turns out that there are four fields that need to be specified, which are denoted in my paper). However a small miracle occurs in three spatial dimensions, in that the divergence-free condition involves only two of the four degrees of freedom (or three out of four, depending on whether one considers a function that is even or odd in to only be half a degree of freedom). This is easiest to illustrate with the scalar NLW (1). Assuming spherical symmetry, this equation becomes

Making the substitution , we can eliminate the lower order term completely to obtain

(This can be compared with the situation in higher dimensions, in which an undesirable zeroth order term shows up.) In particular, if one introduces the null energy density

and the potential energy density

then one can verify the equation

which can be viewed as a transport equation for with forcing term depending on (or vice versa), and is thus quite easy to solve explicitly by choosing one of these fields and then solving for the other. As it turns out, once one is in the supercritical regime , one can solve this equation while giving and the right homogeneity (they have to be homogeneous of order , which is greater than in the supercritical case) and positivity properties, and from this it is possible to prescribe all the other fields one needs to satisfy the conclusions of the main theorem. (It turns out that and will be concentrated near the boundary of the light cone, so this is how the solution will concentrate also.)

I have just uploaded to the arXiv the third installment of my “heatwave” project, entitled “Global regularity of wave maps V. Large data local well-posedness in the energy class“. This (rather technical) paper establishes another of the key ingredients necessary to establish the global existence of smooth wave maps from 2+1-dimensional spacetime to hyperbolic space. Specifically, a large data local well-posedness result is established, constructing a local solution from any initial data with finite (but possibly quite large) energy, and furthermore that the solution depends continuously on the initial data in the energy topology. (This topology was constructed in my previous paper.) Once one has this result, the only remaining task is to show a “Palais-Smale property” for wave maps, in that if singularities form in the wave maps equation, then there exists a non-trivial minimal-energy blowup solution, whose orbit is almost periodic modulo the symmetries of the equation. I anticipate this to the most difficult component of the whole project, and is the subject of the fourth (and hopefully final) installment of this series.

This local result is closely related to the small energy global regularity theory developed in recent years by myself, by Krieger, and by Tataru. In particular, the complicated function spaces used in that paper (which ultimately originate from a precursor paper of Tataru). The main new difficulties here are to extend the small energy theory to large energy (by localising time suitably), and to establish continuous dependence on the data (i.e. two solutions which are initially close in the energy topology, need to stay close in that topology). The former difficulty is in principle manageable by exploiting finite speed of propagation (exploiting the fact (arising from the monotone convergence theorem) that large energy data becomes small energy data at sufficiently small spatial scales), but for technical reasons (having to do with my choice of gauge) I was not able to do this and had to deal with the large energy case directly (and in any case, a genuinely large energy theory is going to be needed to construct the minimal energy blowup solution in the next paper). The latter difficulty is in principle resolvable by adapting the existence theory to differences of solutions, rather than to individual solutions, but the nonlinear choice of gauge adds a rather tedious amount of complexity to the task of making this rigorous. (It may be that simpler gauges, such as the Coulomb gauge, may be usable here, at least in the case of the hyperbolic plane (cf. the work of Krieger), but such gauges cause additional analytic problems as they do not renormalise the nonlinearity as strongly as the caloric gauge. The paper of Tataru establishes these goals, but assumes an isometric embedding of the target manifold into a Euclidean space, which is unfortunately not available for hyperbolic space targets.)

The main technical difficulty that had to be overcome in the paper was that there were two different time variables t, s (one for the wave maps equation and one for the heat flow), and three types of PDE (hyperbolic, parabolic, and ODE) that one has to solve forward in t, forward in s, and backwards in s respectively. In order to close the argument in the large energy case, this necessitated a rather complicated iteration-type scheme, in which one solved for the caloric gauge, established parabolic regularity estimates for that gauge, propagated a “wave-tension field” by the heat flow, and then solved a wave maps type equation using that field as a forcing term. The argument can eventually be closed using mostly “off-the-shelf” function space estimates from previous papers, but is remarkably lengthy, especially when analysing differences of two solutions. (One drawback of using off-the-shelf estimates, though, is that one does not get particularly good control of the solution over extended periods of time; in particular, the spaces used here cannot detect the decay of the solution over extended periods of time (unlike, say, Strichartz spaces for ) and so will not be able to supply the long-time perturbation theory that will be needed in the next paper in this series. I believe I know how to re-engineer these spaces to achieve this, though, and the details should follow in the forthcoming paper.)

More precisely, solutions to (2) tend to decay in time as , as can be seen from the presence of the term in the explicit formula

(4)

for such solutions in terms of the initial position and initial velocity , where , , and dS is the area element of the sphere . (For this post I will ignore the technical issues regarding how smooth the solution has to be in order for the above formula to be valid.) On the other hand, solutions to (3) tend to blow up in finite time from data with positive initial position and initial velocity, even if this data is very small, as can be seen by the family of solutions

for , , and , where c is the positive constant . For T large, this gives a family of solutions which starts out very small at time zero, but still manages to go to infinity in finite time.

The equation (1) can be viewed as a combination of equations (2) and (3) and should thus inherit a mix of the behaviours of both its “parents”. As a general rule, when the initial data of solution is small, one expects the dispersion to “win” and send the solution to zero as , because the nonlinear effects are weak; conversely, when the initial data is large, one expects the nonlinear effects to “win” and cause blowup, or at least large amounts of instability. This division is particularly pronounced when p is large (since then the nonlinearity is very strong for large data and very weak for small data), but not so much for p small (for instance, when p=1, the equation becomes essentially linear, and one can easily show that blowup does not occur from reasonable data.)

The theorem of John formalises this intuition, with a remarkable threshold value for p:

Theorem. Let .

If , then there exist solutions which are arbitrarily small (both in size and in support) and smooth at time zero, but which blow up in finite time.

If , then for every initial data which is sufficiently small in size and support, and sufficiently smooth, one has a global solution (which goes to zero uniformly as ).

[At the critical threshold one also has blowup from arbitrarily small data, as was shown subsequently by Schaeffer.]

The ostensible purpose of this post is to try to explain why the curious exponent should make an appearance here, by sketching out the proof of part 1 of John’s theorem (I will not discuss part 2 here); but another reason I am writing this post is to illustrate how to make quick “back-of-the-envelope” calculations in harmonic analysis and PDE which can obtain the correct numerology for such a problem much faster than a fully rigorous approach. These calculations can be a little tricky to handle properly at first, but with practice they can be done very swiftly.

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